Venture Capital puzzles, solved step by step

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  1. 007A consumer app adds new users equal to 20% of its user base each month and loses 5% of its base to churn each month. Roughly how many months does the user base take to double?Growth and compoundingWarm upConsumer internet VCSeed and early-stage VC

    Try it first

    Answer inside ten seconds.

    Show the worked solution

    About 5 months. Both flows are a share of the same base, so the base grows by 20% minus 5%, a net 15% a month, and that 15% compounds. The rule of 72 gives 72 over 15, about 4.8 months; the exact answer is ln 2 over ln 1.15, 4.96 months. A check: 1.15 to the fifth power is 2.01.

    What is the real growth rate here?

    Picture a water tank with a tap pouring in and a small leak at the bottom. How fast the tank fills depends on the tap minus the leak, not on the tap alone. When new users and lost users are both a share of the same base, the base grows at the difference between the two rates, here 20% minus 5%, or 15% a month. The 20% is the headline the founder will quote; the 15% is the number that moves the base.

    The relationship
    n=ln⁡2ln⁡(1+0.20−0.05)=0.6930.140≈4.96rule of 72: 7215=4.8n = \frac{\ln 2}{\ln(1 + 0.20 - 0.05)} = \frac{0.693}{0.140} \approx 4.96 \qquad \text{rule of 72: } \tfrac{72}{15} = 4.8
    0.20new users each month as a share of the base
    0.05users lost each month as a share of the base
    nmonths for the base to double
    What it says in wordsFind the net monthly rate, then ask how many compounding months turn 1 into 2.
    Growth is what is left after the leak: 15% a month doubles in about 50.51.01.52.02.5double: 2 lakhNet 15%: doubles at 4.96 monthsIgnoring churn (20%): 3.8 monthsaddedlostuser base01234567Months from todayUsers, lakh
    Starting from 1 lakh users, a net 15% monthly rate takes the base past 2 lakh at about 5.0 months, while a curve that ignores the 5% churn would double in 3.8 months; users added and users lost both grow with the base.

    Why is it not 100 divided by 15, about 6.7 months?

    Because each month's 15% is taken on a bigger base than the month before. Adding 15 percentage points a month treats growth as a straight line; compounding means month five adds 15% of 1.75 lakh, not 15% of 1 lakh. The base is 1.75 times its start after four months and 2.01 times after five, so it crosses double just before the end of month five. The rule of 72 gets you to 4.8 in your head, which is close enough to say first and refine.

    What should you say about the assumption?

    Say that you assumed both rates apply to the same opening base each month and stay constant. If churn rises as the app reaches less engaged users, which is common, the doubling time stretches quickly: at 8% churn the net rate is 12% and doubling takes 6.1 months. The flow picture also tells you something the headline does not. Users lost each month grow with the base, so a leak that looks small at 1 lakh users is twice as large in absolute terms at 2 lakh.

    Where candidates lose it

    The quick wrong answer is 3.8 months, taking the 20% acquisition rate as the growth rate. The interviewer gave you the churn figure precisely to see whether you net it off first.

    The quieter loss is the straight-line answer of 6.7 months. Say the rule of 72 out loud, give 4.8, then correct to about 5 with the exact figure; that sequence shows both speed and care.

    What the interviewer asks next

    • If churn rises to 8% a month, how long does doubling take?
    • If new users are a fixed 20,000 a month instead of 20% of the base, where does the base settle?
    • Why might an investor care more about the churn figure than the acquisition figure at this stage?
  2. 081A founder expects to give up 25% of her company in each yearly funding round. How fast must the company's valuation grow each year for the value of her stake to keep growing?Growth and compoundingWarm upSeed and early-stage VCSeries A to C VC

    Try it first

    Quick instinct: what yearly valuation growth keeps her stake's value flat?

    Show the worked solution

    Faster than 33.3% a year; at exactly one third her stake's value stands still. Each round leaves her 75% of her previous ownership, so her stake's value changes by 0.75 times one plus the valuation growth. That equals 1 when growth is 1 over 0.75 minus 1, a third. At 20% growth her stake loses 10% of its value every year, even while the company grows.

    Why is the break-even growth a third and not a quarter?

    A shop sells an item at a 25% discount and wants its revenue back to where it was: it needs a third more units, not a quarter more, because 75 x 1.333 is 100. After giving up a quarter, she holds three quarters, and three quarters of anything must grow by a third to get back to the whole. This is the same asymmetry as recovering from a loss: the recovery is measured on a smaller base, so it needs a bigger percentage.

    With 25% sold each year, the company must grow a third just to stand still50100150200Yr 0Yr 1Yr 2Yr 3Yr 4Yr 520% growth: 5933.3% growth: 10050% growth: 180Value of her stake, start = 100Keep 75% x grow 1.333= 1.00, standing stillFive yearly rounds, 25% sold in each
    Over five yearly rounds that each sell 25%, the founder's stake falls to 59% of its starting value at 20% valuation growth, holds at 100% at 33.3% growth and rises to 180% at 50% growth.
    The relationship
    vt+1vt=(1−d)(1+g)>1  ⟺  g>d1−d=0.250.75=33.3%\frac{v_{t+1}}{v_t} = (1-d)(1+g) > 1 \iff g > \frac{d}{1-d} = \frac{0.25}{0.75} = 33.3\%
    vvalue of her stake, ownership times valuation
    dshare of the company sold each round, 25%
    gyearly growth in valuation
    What it says in wordsHer stake grows only when the valuation grows faster than dilution divided by what she keeps.

    What happens at 20% and at 50% growth?

    At 20%, each year multiplies her stake's value by 0.75 x 1.2 = 0.9. After five rounds the company is worth 2.5 times as much, and her stake is worth only 59% of what it was. At 50%, each year multiplies it by 1.125, and five rounds take it to 1.80 times. Same dilution, opposite outcomes, decided entirely by whether growth clears one third.

    Why would an investor ask this?

    Because it is the test every round must pass for existing holders. With a quarter sold, post-money growth of a third is the same as the new round's pre-money equalling the last round's post-money. A round only makes existing holders richer if the new pre-money beats the old post-money, which is what an up round means. Say that link and the puzzle becomes a point about pricing, not just arithmetic.

    Where candidates lose it

    The instinctive answer is 25%, matching the dilution one for one. It feels symmetric and is wrong, because the growth is applied to what she has left, not to what she had. At 25% growth she still loses about 6% of her value a year.

    The second miss is assuming any growth makes the founder richer. Valuation up and stake value down is common, and naming the condition that separates them is the point of the question.

    What the interviewer asks next

    • At 50% yearly valuation growth, how much can she sell each round and still stand still?
    • Her company's valuation doubles over two rounds, each selling 25%. Is she richer?
    • Why might a founder accept a round that fails this test?